Chapter 6. Processes Resulting from the Intensity-Dependent Refractive Index

Nonlinear Optics Lab . Hanyang Univ.

Chapter 6. Processes Resulting from

the Intensity-Dependent Refractive Index

- Optical phase conjugation - Self-focusing

- Optical bistability - Two-beam coupling - Optical solitons

Reference :

R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.

- Photorefractive effect (Chapter 10)

:

cannot be described by a nonlinear susceptibility c⁽ⁿ⁾for any value of n

Nonlinear Optics Lab . Hanyang Univ.

6.1 Optical Phase Conjugation

: Generation of a time-reversed wavefront

Signal wave :



 











 r k

* s

*

* s

r k s s s

s s

ˆ A E

ˆ A E

i s

i

e e

e e

c.c.

E ) , r ( E ~

s

s

t  e

^it



Phase conjugate wave : E ~ ( r , ) E

_*

c.c.

s

c

t  r e

^it



^where,

r

: amplitude reflection coefficient

Nonlinear Optics Lab . Hanyang Univ.

r k

* s

*

* s

A

s

E  e ˆ

_s

e

^{i }

Properties of phase conjugate wave :

1)

e ˆ

^*_s : The polarization state of circular polarized light does not change in reflection from PCM Ex)

eˆ

_s

 e

₀

i  e

₀

e

^i/2

j

i) In reflection from metallic mirror [-phase shift]

ii) In reflection from PCM [-phase shift &

y component : i/2  -i/2 (- : delayed)]

2)

A

^*_s : The wavefront is reversed

) r

* ( )

r

( _{t t}

A

A

_s

 a

_s

e

^i



_s

 a

_s

e

^i

3)

e

^iksr : The incident wave is reflected back into its direction of incidence

s

s

k

k  

Nonlinear Optics Lab . Hanyang Univ.

Aberration correction by optical phase conjugation

Wave equation :

0 ) ~

r (

~

2 2 2

2





 

 t

E E  c

Solution : ~ ( r , ) ( r )

_{( )}

. . c c e

A t

E 

^{i kzt}



With slow varying approximation, ( r )

₂

2 0

2 2

2





 

 

 



 



 z

ik A A c k

T

A





Since the equation is generally valid, so is its complex conjugate, which is given explicitly by 0

) 2 r

(

_{2 *} ^*

2 2

*

2





 

 

 



 



 z

ik A A

c k

T

A





Solution : ~ ( r , )

_*

( r )

_{( )}

. . c c e

A t

E

_c



^{i kzt}



: A wave propagating in the –z direction whose complex amplitude is everywhere the complex of the forward-going wave

Nonlinear Optics Lab . Hanyang Univ.

Nonlinear Optics Lab . Hanyang Univ.

Phase Conjugation by Degenerated Four-Wave Mixing

1) Qualitative understanding

Four interacting waves :

~ ( r , ) ( r ) . . ( r )

_{( r )}

. . c c e

A c c e

E t

E

_i



_i ^it

 

_i ^{i ki t}



^{(i1,2,3,4)}

Nonlinear polarization :

P

^NL

 6 c

⁽³⁾

E

₁

E

₂

E

₃^*

 6 c

⁽³⁾

A

₁

A

₂

A

₃^*

e

^{i(k1k2k3)r}

0(

2

1k  

k Counter-propagating waves)

r

* 3 2 1 ) 3

( ₃

6

^{ }

 c A A A e

^ik

New wave (k₄) source term !

So, A

₄

is proportional to A

₃^*

(complex conjugate of A

₃

)

and its propagation direction is –k

₃

(in the case of perfect phase matching)

Nonlinear Optics Lab . Hanyang Univ.

2) Rigorous treatment

Total field amplitude : E  E

₁

 E

₂

 E

₃

 E

₄

Nonlinear polarization : P

^NL

 3 c

⁽³⁾

(        ) E

²

E

^*



3 4 2^*



* 4 4 1

* 3 3 1

* 2 2 1

* 1 2 1 ) 3 (

1

3 E E 2 E E E 2 E E E 2 E E E 2 E E E

P     

 c



₂² ₂^* _{2 1 1}^* _{2 3 3}^* _{2 4 4}^* _{3 4 1}^*



) 3 (

2

3 E E 2 E E E 2 E E E 2 E E E 2 E E E

P  c    



1 2 4^*



* 4 4 3

* 2 2 3

* 1 1 3

* 3 2 3 ) 3 (

3

3 E E 2 E E E 2 E E E 2 E E E 2 E E E

P  c    



₄² ₄^* _{4 1 1}^* _{4 2 2}^* _{4 3 3}^* _{1 2 3}^*



) 3 (

4

3 E E 2 E E E 2 E E E 2 E E E 2 E E E

P  c    

4 3 2

1

, E E , E

E 

 Neglect the 2^nd order terms of E₃ and E₄



1 2 2^*



* 1 2 1 ) 3 (

1

3 E E 2 E E E

P  c 



2 1 1^*



* 2 2 2 ) 3 (

2

3 E E 2 E E E

P  c 



_{3 1 1}^* _{3 2 2}^* _{1 2 4}^*



) 3 (

3

3 2 E E E 2 E E E 2 E E E

P  c  



_{4 1 1}^* _{4 2 2}^* _{1 2 3}^*



) 3 (

4

3 2 E E E 2 E E E 2 E E E

P  c  

Nonlinear Optics Lab . Hanyang Univ.

Wave equation :

i i

i

P

t c t

E

E c ~ 4 ~

~

2 2 2 2

2 2 2



 



 

  

where,

~ ( r , ) ( r ) . . ( r )

_{( r )}

. . c c e

A c c e

E t

E

_i



_i ^it

 

_i ^{i kit}



(1) Pump waves, A

₁

and A

₂

(slow varying approximation)

 

1 1 1

2 2 2

1 ) 3

1

6

(

| | 2 | |

A i A A nc A

i dz

dA _   c _{ } 



 

2 2 2

2 1 2

2 ) 3 2 (

|

| 2

| 6 |

A i A A nc A

i dz

dA _{ }   c _{  } 

# Each wave shifts the phase of the other wave by twice as much as it shift its own phase

# Since only the phase of the pump waves are affected by nonlinear coupling, the quantities

|A₁|² and |A₂|² are spatially invariant, and hence the k1 and k2 are in fact constant

Solution :

z

e

i

A z

A

₁

( ) 

₁

( 0 )

^1

A

₂

( z )  A

₂

( 0 ) e

^i2z

* 3 ) ( 2

1

* 3 2 1 4

2

)

1

0 ( ) 0

( A e A

A A A A

P  

^{i   z}

(6.1.15)

: Nonlinear polarization responsible for producing the phase conjugate wave varies spatially.

) 0 ( ) 0 ( ) ( ) ( )

0 ( )

0

(

_{2 1 2 1 2 1 2}

1

A A z A z A A

A       

Therefore, two pump beams should have equal intensities :

Nonlinear Optics Lab . Hanyang Univ.

(2) Signal ( A

₃

) and conjugate waves ( A

₄

)

 

3 3 4^*

* 4 2 1 3 2 2 2

1 ) 3 3 (

)

|

| 2

| 12 (|

A i A i A A A A A

nc A i dz

dA _   c _{  }  _ 

 

3 4 3^*

* 3 2 1 4 2 2 2

1 ) 3

4

12

(

(| | 2 | | )

A i A i A

A A A A

nc A i dz

dA _{ }   c _{   }  _ 

where,

12 (| | 2 | |

₂

)

2 2

1 ) 3 (

3

A A

nc

i 

   c



2 1 ) 3

12

(

A nc A

i  c

  

put,

A

₃

 A

₃^'

e

^i3z

z

e

i

A A

₄



₄^{' 4}

' 3 '

4

' 4 '

3

A dz i

dA

A dz i

dA













) 4 , 3

0

(

|

|

^{2 '}

2 '









ⁱ

A

_i i

dz

dA 

Nonlinear Optics Lab . Hanyang Univ.

Solution :

) 0

| (

| cos

)]

(

| sin[|

) |

| (

| cos

|

| ) cos (

) 0

| (

| cos

)]

(

| cos[|

)

| (

| cos

|

| sin

| ) |

(

'*

3 '

4 '

4

'*

3 '

4 '*

3

L A L z L i

L A z z

A

L A L L z

L A z z i

A

























 



 





0 )

'

(

4

L 

A

( conjugate wave at z=L is zero)

) 0 ( )

|

|

| (tan ) |

0 (

|

| )/cos 0 ( )

(

'*

3 '

4

'*

3 '*

3

A i L

A

L A

L A

 









i)

A

₃^'*

( L )  A

₃^'*

( 0 )



 0 ~ ) 0

'

( A

4

ii)

: amplification

: depends on

|  | L

(can exceed 100%  pump wave energy)

Nonlinear Optics Lab . Hanyang Univ.

Processes of degenerated four-wave mixing

:

One photon from each of the pump waves is annihilated

and one photon is added to each of the signal and conjugate waves

one photon transition two photon transition wave-vectors

 Amplification of A₃ and over 100% conversion of A₄/

A

₃ are possible

Nonlinear Optics Lab . Hanyang Univ.

Experimental set-ups

0 )

(k₁k₂  (k₃k₄)0

A1

A2

A3

A4

4

3 A

A 

Nonlinear Optics Lab . Hanyang Univ.

17.5 Optical Resonator with Phase Conjugate Reflectors (A. Yariv)

1 8

9

1 7

8

1 6

7

1 5

6

1 4

5

1 3

4

1 2

3 1 2

) , (

) , (

) , ( )

, (

) , (

)) , ( (

) , ( )

, (





































































































































































n m

n m

n m n

m n m

n m

n m n

m

l

l

l l

R

R l

l R

l R l

R

R



 

_l

^{( m , n )}

# The self-consistence condition is satisfied automatically every two round trips.

 The phase conjugate resonator is stable regardless of the radius of curvature R of the mirror and the spacing l.

Nonlinear Optics Lab . Hanyang Univ.

17.6 The ABCD Formalism of Phase Conjugate Optical Resonator

The wave incident upon the PCM :

 

 



  

 



 



   

 )

( 2 exp ) ( 2 )

( exp ) ( E

2 2

2 2

i i

i

i

q

kz kr t ε i

w r kz kr

t

ε i 

  r

r

 

 



  

 



 



   

 )

( 2 exp ) ( 2 )

( exp ) ( E

2

* 2 2 2

*

r i

i

r

q

kz kr t ε i

w r kz kr

t

ε i 

  r

r

2

1 1

w - i q

_i





 

where,

Reflected conjugate wave :

* 2

1 1

1

i

r

w q

i

q     

 





 

 



 

 

 



 



 

 0 - 1

0 1 D

C

B , A

D C

B A

*

*

M

i i

r

q

q q

Ray transfer matrix for the PCM mirror

By comparing the ABCD law for ordinary optical elements,

Nonlinear Optics Lab . Hanyang Univ.

ABCD law at any plane following the PCM :

T

* T

T

* T

D C

B A



 

i i

out

q

q q

Example)



 





 















 











 



 





 



 



























 





1 0

0 1 R 1

2 -

0 1

D C

B A 1 0

0 1 D C

B A R 1

2 -

0 1 D

C B A

1 1

1 1

M1

Matrix after one round trip :

Matrix after two round trip :

 

M I

M 

 







 





 















 





 













 0 1

0 1 1 0

0 1 R 1

2 -

0 1 1 0

0 1 R 1

2 -

0

2 1

1 2

: Self-consistence condition is satisfied automatically every two round trips

Nonlinear Optics Lab . Hanyang Univ.

17.7 Dynamic Distortion Correction within a Laser Resonator

Phase conjugate resonator

Distortion corrected beam

Nonlinear Optics Lab . Hanyang Univ.

17.8 Holographic Analogs of Phase Conjugate Optics

1) Holography

recording reading

Nonlinear Optics Lab . Hanyang Univ.

2) Phase conjugate optics

4

3 A

A 

Holography by phase conjugation

- Real time processing (no developing process) - Distortion free image transmission

Nonlinear Optics Lab . Hanyang Univ.

17.9 Imaging through a Distorted Medium

Distortion free transmission (A₂)

Nonlinear Optics Lab . Hanyang Univ.

6.2 Self-Focusing of Light

2

I

0

n

n n  

2

e

2

)

I( r 

^{r w}

Gaussian beam :

 







defocusing -

self

focusing -

self

: 0

: 0

2 2

n

n

Nonlinear Optics Lab . Hanyang Univ.

Self-Trapping

: Beam spread due to diffraction is precisely compensated by the contraction due to self-focusing

Simple model for self-trapping

Critical angle for total internal reflection :

n n

n

 

 

0 0

cos

0 0

 0



2 1

0 0

0 2

0

2 2 1 1 1

 

 



 











n n n

n

 

 

Nonlinear Optics Lab . Hanyang Univ.

A laser beam of diameter d will contain rays within a cone whose maximum angular extent is of the order of magnitude of diffraction angle ;

d n d

d

n

0 0

6 .

2 0 



   

So, the condition for self-trapping :



0



_d 

2

0 0

0 2

1

0

6 . 0 2 6 1

. 2 0



 



 



 



 



 

n dn d n

n n

n

  



 ²

_{0 2}

^I 

¹²

6 .

0

^



 d  n n

2

I n n



Critical laser power :

 

2 0

2 2

8 6 . I 0

P 4

n d n

cr





 _



# Independent of the beam diameter

Ex) CS₂, n₂=2.6x10^-14 cm²/W, n₀=1.7, =1m

 P_cr = 33 kW

Nonlinear Optics Lab . Hanyang Univ.

Simple model of self-focusing

2w

₀

z

_f

n n



2 1

0 n n₀



2 ) ( 1

) (

)

(n₀ n z² w₀^{2 12} n₀ n z_f

  

_f

  

2 0 0 0

0 2

1 2

) 1

(

 





 



 







f f

f f

f z

n w z n

z n

z n n

z

 

2 1

0 0

0 0

2 1 0

0

2

2       



 

 

 





n n w

n w n

z

_f

 





where, : critical angle

2I nn



12

2 2 0 0 12

2 0

0 I 2

 

 



 

 

 



 



n P

w n n

w n

z_f



where, I

2 1 ₂

w0

P

 

: total power

 

¹²

2 0

0 1

6 . 0 2

cr

f P P

w z n

 

Nonlinear Optics Lab . Hanyang Univ.

6.3 Optical Bistability

: Two different output intensities for a given input intensity

 Switch in optical computing and in optical computing

Bistability in a nonlinear medium inside of a Fabry-Perot resonator

T

R 



²

2

, 



Intensity reflectance and transmittance :



 ,

where, : amplitude reflectance and transmittance

l

e

ikl ^



^



₂

 A

₂ ²

A

2 1

2

A A

A     

l

e

ikl ^













_{2 2 2}¹

1

A A

^(6.3.3)

Nonlinear Optics Lab . Hanyang Univ.

1) Absorptive Bistability

In the case when only the absorption coefficient depends nonlinearly on the field intensity, at the resonance condition,



²

e

^2ikl

 R

Assume,

 l  1

) 1

( 1

A

₂

A

¹

l R 





 

  ^

 2 A

²

I

_i

nc 

_i



¹



2 2

I I

1 (1 )

T

R  l

  

Introducing the dimensionless parameter C,

R l C R

  1



2 1

2

( 1 )

I I 1

C T 



^(6.3.7)

Nonlinear Optics Lab . Hanyang Univ.

Assume the absorption coefficient obeys the relation valid for a two-level saturable absorber ;

I

S

I 1

0

  



Intracavity intensity :

I

₂

 I

^'₂

 2 I

₂

) 1 , (

I 2I 1

0 0 2

0

R l

C R C C

S

 

 



_where,



2

2 0 2

1

I 1 1 2I I

I 



 





 



S

T C

(6.3.7) 

2

3

I

I  T

Nonlinear Optics Lab . Hanyang Univ.

2) Dispersive Bistability

In the case when only the refractive index depends nonlinearly on the field intensity,

) I ( ,

0 n  f

 

(6.3.3)  _{ikl iδ}

e R e  

 

1 A 1

A

₂

A

₂¹₂



¹





^where,



²

 R e

ⁱ

2

0





  

c l n







₀

 

2 ₀ : linear phase shift : nonlinear phase shift cl

n





₂2 ₂I

 ⁴  ^{sin   2}

1

I

) cos 2

1 (

I )

1 )(

1 ( I I

2 2

1

2 1 1

2



 T

R

T

R R

T e

R e

R T

iδ iδ

 



 



 

_

Similarly as before,

Nonlinear Optics Lab . Hanyang Univ.

 ⁴  ^{sin   2}

1

1 I

I

2 2

1

2

R T 

T

 

2 2

0

4  I



 



 

 l

n C 





2

3

I

I  T